How to solve the indefinite integral of the function y=/x*square root[(ln x)^2-1]?

To solve the indefinite integral, we'll replace ln x by
t:

ln x = t

We'll
differentiate both sides:

dx/x =
dt

We'll re-write the integral in the new variable
t:

Int dx/x*sqrt[(lnx)^2 - 1] = Int
dt/sqrt(t^2-1)

Int dt/sqrt(t^2-1) = ln|t + sqrt(t^2-1)| +
C

The indefinite integral is: Int
dx/x*sqrt[(lnx)^2 - 1] = ln|ln x + sqrt[(lnx)^2 - 1]| +
C